7.7-7.20【Xin Nie】Higgs bundles, higher Teichmüller theory and minimal surfaces (I, II, III, IV)

Time：2022-07-01Views：56

Speaker: Xin Nie (Southeast University)

Time: 09:00-10:30, Jul 7,8,19,20, 2022

Place: Room 5106, the Fifth Teaching Building

1. Holomorphic vector bundles and Higgs bundle

• Symplectic reduction and its Kähler/hyperkähler versions

• Relation with G.I.T. (Kempf-Ness theorem)

• Holomorphic vector bundles, Atiyah-Bott's reduction and Narasimhan-Seshadri theorem

• Higgs bundles, Hitchin's reduction and Hitchin-Kobayashi correspondence

2. G-Higgs bundles and surface group representations

• SL(2,R)-Hitchin component and Teichmüller theory

• SL(n,R)-Hitchin component, cyclic Higgs bundles

• Theory of semsimple Lie algebras and principal 3d subalgebras

• G-Higgs bundles for a general Lie group G

• SO(p,q)-Higgs bundles

3. Harmonic maps associated to G-Higg bundles and Labourie's conjecture

• harmonic maps, minimal surfaces and Riemannian symmetric spaces

• harmonic maps given by G-Higgs bundles

• Energy functional and Labourie's conjecture

• Strategy via Infinitesimal rigidity and the SL(3,R)-case (affine spheres)

• Cyclic surfaces (Labourie Ann.Math.2017)

4. Minimal surfaces in pseudo-hyperbolic spaces

• pseudo-hyperbolic spaces

• second variation formula and maximal surfaces

• Relation with Teichmüller theory (Bonsante-Schlenker Invent.Math.2010)

• Labourie's conjecture for SO(2,n) (Collier-Tholozan-Toulisse Duke.Math.J. 2019)

• A-surfaces and their infinitesimal rigidity (speaker arXiv:2206.13357)